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A magician has five animals in his magic...

A magician has five animals in his magic hat : 3 doves and 2 rabbits. If he pulls two animals out of the hat at random, what is the chance that he will have a matched pair?

A

50%

B

40%

C

60%

D

30%

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of finding the probability that the magician pulls out a matched pair of animals (either both doves or both rabbits), we can follow these steps: ### Step 1: Identify the total number of animals The magician has a total of 5 animals: 3 doves and 2 rabbits. ### Step 2: Calculate the total number of ways to choose 2 animals from 5 The total number of ways to choose 2 animals from 5 can be calculated using the combination formula: \[ \text{Total ways} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of animals and \( r \) is the number of animals to choose. For our case: \[ \text{Total ways} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \] ### Step 3: Calculate the number of ways to pick a matched pair of rabbits To find the number of ways to pick 2 rabbits from the 2 available: \[ \text{Ways to choose 2 rabbits} = \binom{2}{2} = 1 \] ### Step 4: Calculate the number of ways to pick a matched pair of doves To find the number of ways to pick 2 doves from the 3 available: \[ \text{Ways to choose 2 doves} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \] ### Step 5: Calculate the total number of ways to get a matched pair The total number of ways to get a matched pair (either both rabbits or both doves) is: \[ \text{Total matched pairs} = \text{Ways to choose 2 rabbits} + \text{Ways to choose 2 doves} = 1 + 3 = 4 \] ### Step 6: Calculate the probability of getting a matched pair The probability of getting a matched pair is given by the ratio of the number of favorable outcomes to the total outcomes: \[ P(\text{matched pair}) = \frac{\text{Total matched pairs}}{\text{Total ways}} = \frac{4}{10} = \frac{2}{5} \] ### Final Answer The probability that the magician will pull out a matched pair of animals is \( \frac{2}{5} \) or 0.4. ---
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